1980 AMC 12 第 10 题

Question

三个互相啮合的圆形齿轮

A

、

B

、

C

的齿数分别为

x

、

y

、

z

。（所有齿轮的齿大小相同，并且如图所示均匀排列。）

A

、

B

、

C

的角速度（单位为每分钟转数）之比为

The number of teeth in three meshed circular gears

A,

B,

C

are

x,

y,

z,

respectively. (The teeth on all gears are the same size and regularly spaced as in the figure.) The angular speeds, in revolutions per minute, of

A,

B,

C

are in the proportion

x:y:z

z:y:x

y:z:x

yz:xz:xy

xz:yx:zy

齿轮啮合处的切向速度大小相同

Meshed teeth have the same tangential speed at their points of contact

角速度与圆周长成反比，因而与齿数成反比

Angular speed is inversely proportional to circumference and hence to the number of teeth

各齿轮的圆周长分别与

x

、

y

、

z

成正比，而它们的切向速度大小相等。因此它们的角速度之比为

\frac1x:\frac1y:\frac1z\text{。}

三项同时乘以

xyz

，得到

yz:xz:xy

。

因此，正确答案是 D。

The gears’ circumferences are proportional to

x,

y,

and

z,

while their tangential speeds have equal magnitude. Their angular speeds are therefore proportional to

\frac1x:\frac1y:\frac1z.

Multiplying all three terms by

xyz

gives

yz:xz:xy.

Therefore, the correct answer is D.

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